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Variable Metric Forward–Backward Algorithm for Minimizing the Sum of a Differentiable Function and a Convex Function

Author

Listed:
  • Emilie Chouzenoux

    (Université Paris-Est Marne-la-Vallée)

  • Jean-Christophe Pesquet

    (Université Paris-Est Marne-la-Vallée)

  • Audrey Repetti

    (Université Paris-Est Marne-la-Vallée)

Abstract

We consider the minimization of a function G defined on ${ \mathbb{R} } ^{N}$ , which is the sum of a (not necessarily convex) differentiable function and a (not necessarily differentiable) convex function. Moreover, we assume that G satisfies the Kurdyka–Łojasiewicz property. Such a problem can be solved with the Forward–Backward algorithm. However, the latter algorithm may suffer from slow convergence. We propose an acceleration strategy based on the use of variable metrics and of the Majorize–Minimize principle. We give conditions under which the sequence generated by the resulting Variable Metric Forward–Backward algorithm converges to a critical point of G. Numerical results illustrate the performance of the proposed algorithm in an image reconstruction application.

Suggested Citation

  • Emilie Chouzenoux & Jean-Christophe Pesquet & Audrey Repetti, 2014. "Variable Metric Forward–Backward Algorithm for Minimizing the Sum of a Differentiable Function and a Convex Function," Journal of Optimization Theory and Applications, Springer, vol. 162(1), pages 107-132, July.
    • Handle: RePEc:spr:joptap:v:162:y:2014:i:1:d:10.1007_s10957-013-0465-7
    DOI: 10.1007/s10957-013-0465-7
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    References listed on IDEAS

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    1. Hunter D.R. & Lange K., 2004. "A Tutorial on MM Algorithms," The American Statistician, American Statistical Association, vol. 58, pages 30-37, February.
    2. NESTEROV, Yu., 2007. "Gradient methods for minimizing composite objective function," LIDAM Discussion Papers CORE 2007076, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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    Cited by:

    1. Sixuan Bai & Minghua Li & Chengwu Lu & Daoli Zhu & Sien Deng, 2022. "The Equivalence of Three Types of Error Bounds for Weakly and Approximately Convex Functions," Journal of Optimization Theory and Applications, Springer, vol. 194(1), pages 220-245, July.
    2. Bonettini, S. & Prato, M. & Rebegoldi, S., 2021. "New convergence results for the inexact variable metric forward–backward method," Applied Mathematics and Computation, Elsevier, vol. 392(C).
    3. Peter Ochs, 2018. "Local Convergence of the Heavy-Ball Method and iPiano for Non-convex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 177(1), pages 153-180, April.
    4. Daoli Zhu & Sien Deng & Minghua Li & Lei Zhao, 2021. "Level-Set Subdifferential Error Bounds and Linear Convergence of Bregman Proximal Gradient Method," Journal of Optimization Theory and Applications, Springer, vol. 189(3), pages 889-918, June.
    5. Silvia Bonettini & Peter Ochs & Marco Prato & Simone Rebegoldi, 2023. "An abstract convergence framework with application to inertial inexact forward–backward methods," Computational Optimization and Applications, Springer, vol. 84(2), pages 319-362, March.
    6. S. Bonettini & M. Prato & S. Rebegoldi, 2018. "A block coordinate variable metric linesearch based proximal gradient method," Computational Optimization and Applications, Springer, vol. 71(1), pages 5-52, September.
    7. Emilie Chouzenoux & Jean-Christophe Pesquet & Audrey Repetti, 2016. "A block coordinate variable metric forward–backward algorithm," Journal of Global Optimization, Springer, vol. 66(3), pages 457-485, November.
    8. Szilárd Csaba László, 2023. "A Forward–Backward Algorithm With Different Inertial Terms for Structured Non-Convex Minimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 198(1), pages 387-427, July.
    9. Radu Ioan Boţ & Ernö Robert Csetnek & Szilárd Csaba László, 2016. "An inertial forward–backward algorithm for the minimization of the sum of two nonconvex functions," EURO Journal on Computational Optimization, Springer;EURO - The Association of European Operational Research Societies, vol. 4(1), pages 3-25, February.
    10. Ching-pei Lee & Stephen J. Wright, 2019. "Inexact Successive quadratic approximation for regularized optimization," Computational Optimization and Applications, Springer, vol. 72(3), pages 641-674, April.
    11. Tianxiang Liu & Akiko Takeda, 2022. "An inexact successive quadratic approximation method for a class of difference-of-convex optimization problems," Computational Optimization and Applications, Springer, vol. 82(1), pages 141-173, May.
    12. Radu Ioan Boţ & Ernö Robert Csetnek, 2016. "An Inertial Tseng’s Type Proximal Algorithm for Nonsmooth and Nonconvex Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 171(2), pages 600-616, November.
    13. J. C. De Los Reyes & E. Loayza & P. Merino, 2017. "Second-order orthant-based methods with enriched Hessian information for sparse $$\ell _1$$ ℓ 1 -optimization," Computational Optimization and Applications, Springer, vol. 67(2), pages 225-258, June.

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